Why Apollo missions 2 - 6 have no patches?

Officially, there was no Apollo 2 or 3. These were unmanned flights that took place after the Apollo 1 fire, while officials were still trying to decide if they wanted to continue using the Apollo name. By the time they had made up their minds they had already launched 2 test flights of the Saturn V. Following after those, they flew three more unmanned tests, named Apollo 4, 5, and 6. The first manned mission was Apollo 7.

Actually, that's not quite it. First of all, patches were commissioned only for manned missions, and contrary to the Wikipedia article, examination of the photographs shot during training reveal that the patch Grissom, White and Chaffee designed for AS-204 has no designation at all. Second, the unmanned flights that would have been designated Apollo 1, Apollo 2, and Apollo 3 flew BEFORE the flight that Betty Grissom insisted be designated Apollo 1. AS-201 (Apollo 1) flew on 26 February 1966, AS203 (Apollo 3) flew on 5 July 1966, and AS-202 (Apollo 2) flew on 25 August 1966. The flights after AS-204 used the normal designations assigned after liftoff. Note that, by this scheme, AS-204, had it been launched, would have been designated Apollo 4, not Apollo 1.